Stationary measures can also be seen as part of invariant measures on skew-product systems. Let \(\hat{Y}\)  denote the product \(H^\mathbb {Z}\times Y\)  and \(\hat{T}\)  the skew-shift given by \(\hat{T}((b,a),y)=(T(b,a), a_0 y)\) . A basic fact (see e.g.  [1]}) is that given a probability measure \(\mu \)  on \(H\) , any \(\mu \) -stationary measure on \(Y\)  gives rise to a \(\hat{T}\) -invariant measure on \(\hat{Y}\)  that projects onto \(\mu ^\mathbb {Z}\)  on the \(H^\mathbb {Z}\)  factor: indeed given a \(\mu \) -stationary measure \(\nu \) , the measure
\(\hat{\nu }=\int \delta _{w} \otimes \nu _{w} d\mu ^\mathbb {Z}(w)\) 
